jmc

Consider the triangle with angle 4; label the other two angles angle 5 and angle 6. By triangle angle-sum, $m\angle 1+m\angle 2+m\angle 3+m\angle 5+m\angle 6=180^{\circ }$, or $$m\angle 5+m\angle 6=180^{\circ }-76^{\circ }-27^{\circ }-17^{\circ }=60^{\circ }$$ By triangle angle-sum on the small triangle, $m\angle 4+m\angle 5+m\angle 6=180^{\circ }$, so $60^{\circ }+m\angle 4=180^{\circ }$ or $m\angle 4=\boxed{120^{\circ }}$.